Ex. 2. Required the sum of 10 terms of the series 12, 9, 6, &c. In this example, a 12, 6-3, and n=10, and, there 10 fore, s = (24+(10—1) × —3) = (24—27)5 — —8 × 15 — X = -15. Ex. 3. The sum of 8 terms of an arithmetical progression is 105, and the first term is 3, required the common difference. By substitution in the fundamental equation, we have 7 = 105, that is, 6+66= (6+(7—1)b) 1/2: 210 30, or, 6b = EXERCISES. Ex. 1. Required the sum of 8 terms of the series 2, 5, 8, &c. Ans. 100. Ex. 2. Required the sum of 12 terms of the series, 1, 11, 2, &c. Ans. 39. Ex. 3. The sum of an arithmetic series is 440, first term 3, and common difference 2. What is the number of terms? Ans. 20. Ex. 4. If 100 stones be placed on the ground in a straight line, at the distance of a yard from each other, how far will a person travel, who shall bring them one by one to a basket, placed at the distance of a yard from the first stone? Ans. 5 miles and 1300 yards Ex. 5. A, travels uniformly at the rate of 6 miles an hour, and set off upon his journey 3 hours and 20 minutes before B; B follows him at the rate of 5 miles the first hour, 6 the second, 7 the third, and so on. In how many hours will B overtake A ? Ans. In 8 hours. ON GEOMETRICAL PROGRESSION. 129. When a series of quantities is formed from the continual multiplication of the same quantity, by the same multiplier, these quantities are said to be in geometrical progression. Thus, 1, 2, 4, 8, &c,; 1, §, 1, †, &c., are quantities in geometrical progression. The constant multiplier is called the common ratio, and is manifestly equal to the quotient arising from the division of any term, by that which immediately precedes it. 130. If a be the first term, and r the common ratio of a series of terms in geometrical progression, then the second term will be ar, the third ar2, the fourth ar3, &c., or, since the index of r in any term is less by unity than the number of terms, reckoning from the beginning, the last term will be ar2-1, when n denotes the number of terms. Hence, if / denote the last term, we shall have l = arn-1. 131. The sum, of a series of quantities in geometrical progression, may be found, by subtracting the first term from the product of the last term and common ratio, and dividing the remainder by the difference between the common ratio and unity. Hence, (r-1)sar"—a, and, therefore, s = lar, therefore, rlar", and hence, by substitution, s = Fla If r is a proper fraction, then r and its powers are less than unity, and, consequently, both numerator and denominator of the fraction rl-a would become negative. The division of the one by the other would no doubt give a positive result, but it will be rather more convenient to transform the equation s = rl-a into the form s= -rt 132. When r is a proper fraction, and n indefinitely great, ar" becomes indefinitely small; and, therefore, s = a the limit to which the sum of the series approaches. 133. From the equation s = quantities 8, r, and a 7 and a found. rl-a r any three of the four r being given, the fourth may be &c. EXAMPLES. Ex. 1. Required the sum of 8 terms of the series 1, 2, 4, 8, Ex. 2. Required the sum of 6 terms of the series, 1, 1, 1. &c. When n is a large number, the solution of the fundamental equation is most conveniently effected by Logarithms, as will afterwards appear. Ex. 3. Required the value of the repeating decimal .3333. 3 This decimal is represented by the geometric series + 10 3 3 + + &c., whose first term is and common ratio 100 1000 10' 3 1 10' Ex. 1. Required the sum of the series 1, 2, 4, 8, 16, &c. to 14 terms. Ans. 16383. Ex. 2. Find the value of the series, 1, }, }), 27, &c. continued indefinitely. Ans.. Ex. 3. Required the value of the circulating decimal 0.45. Ans. T Ex. 4. A person sold a horse, on condition of his receiving a farthing for the first nail in his shoes, a halfpenny for the second, a penny for the third, and so on, doubling the price of every nail, to 32, the number of nails in the four shoes. Required the price of the horse. Ans. L. 4,473,924, 5s. 3 d. ELEMENTS OF GEOMETRY. BOOK I. DEFINITIONS. I. A POINT is that which has position but not magni- Book I. tude. II.-A line is length without breadth. Corollary. The extremities of a line are points; and the intersections of one line with another are also points. III.-If two lines are such, that they cannot coincide in any two points, without coinciding altogether, each of them is called a straight line. Cor.-Hence, two straight lines cannot inclose a space. Neither can two straight lines have a common segment; for they cannot coincide in part, without coinciding altogether. IV. A superficies is that which has only length and breadth. Cor. The extremities of a superficies are lines, and the intersections of one superficies with another, are also lines. V.-A plane superficies is that in which any two points being taken, the straight line between them lies wholly in that superficies. K |